The object of this paper, which is the first in a series of three, is to lay the foundations of the theory of ideals and algebraic sets over groups.

We re-cast in a more combinatorial and computational form the topological approach of J. Stallings to the study of subgroups of free groups.  2002 Elsevier Science (USA)

We prove that any two non-abelian free groups have the same elementary theory and that this theory is decidable. These results solve two questions that were raised by Tarski in 1945.

We give a precise definition of "generic-case complexity" and show that for a very large class of finitely generated groups the classical decision problems of group theory -the word, conjugacy and membership problems -all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs.

We introduce the notion of a regular quadratic equation and a regular NTQ system over a free group. We prove the results that can be described as implicit function theorems for algebraic varieties corresponding to regular quadratic and NTQ systems. We will also show that the implicit function theorem is true only for these varieties. In algebraic geometry such results would be described as lifting solutions of equations into generic points. From the model theoretic view-point we claim the existence of simple Skolem functions for particular ∀∃-formulas over free groups. Proving these theorems we describe in details a new version of the Makanin-Razborov process for solving equations in free groups. We also prove a weak version of the implicit function theorem for NTQ systems which is one of the key results in the solution of the Tarski's problems about the elementary theory of a free group. We call it the parametrization theorem.  2005 Elsevier Inc. All rights reserved.